1. Introduction
The notion of uniformity has been investigated by several mathematician as Weil [2] - [4] , Cohen [5] [6] , and Graves [7] .
The theory of uniform spaces was given by Burbaki in [8] . Also Wiels in his booklet [4] defined uniformly continuous mapping. For more information about Uniform spaces one my refer to [9] .
In 2009, Tallafha, A. and Khalil, R. [10] , defined a new type of uniform spaces, namely semi-linear uniform spaces and they gave example of semi-linear space which was not metrizable. Also they defined a set valued map
on
, by which they studied some cases of best approximation in such spaces. More precisely, they gave the following.
Let
be a semi-linear uniform space;
is proximinal if for any
, there exists
such that
. They asked that “must every compact is proximinal”, they gave the answer for the cases―i) E is finiate; ii) If
converges to x, then
is proximinal.
In [11] , Tallafha, A. defined another set valued map
on
, and gave some properties of semi-linear uniform spaces using the maps
and
. Also in [1] [12] , Tallafha defined Lipschitz condition and con- tractions for functions on semi-linear uniform spaces, which enabled us to study fixed point for such functions. Lipschitz condition, and contractions are usually discussed in metric and normed spaces and never been studied in other weaker spaces. We believe that the structure of semi-linear uniform spaces is very rich, and all the known results on fixed point theory can be generalized.
The object of this paper is to generalize the definition of Lipschitz condition, and contraction mapping on semi-linear uniform spaces given by Tallafha [12] . Also we shall give a new topopological properties and more properties of semi-linear uniform spaces.
2. Semi-Linear Uniform Space
Let X be a set and
be a collection of subsets of
, such that each element V of
contains the diagonal
, and
for all
(symmetric).
is called the family of all entourages of the diagonal.
Definition 1 [10] . Let
be a sub collection of
, the pair
is called a semi-linear uniform space if,
i)
is a chain.
ii) For every
, there exists
such that
.
iii)
.
iv)
.
Definition 2 [10] . Let
be a semi-linear uniform space, for
, let
. Then, the set valued map
on
is defined by ![]()
Clearly for all
, we have
and
. Let
, from now on, we shall denote
by
.
Definition 3 [11] . Let
be a semi-linear uniform space. Then, the set valued map
on
is
defined by, ![]()
The following results are given in [12] .
Proposition 1. Let
be a semi-linear uniform space, and
is a sub collection of
, then
, if an only if there exist
such that
.
Corollary 1. Let
be a semi-linear uniform space. If
, then,
1) There exist
such that
.
2)
.
Let
the family of all entourages of the diagonal, then for all
, by nV, we mean
n-times and
, so for all
,
and
for all
.
Proposition 2. Let
be a semi-linear uniform space. If
, then
.
Proposition 3. Let
be a semi-linear uniform space. If
is a sub collection of
, then
.
Question. Does
?
3. Topological Properties of Semi-Linear Uniform Spaces
Definition 4 [13] . For
and
. The open ball of center x and radius V is defined by
, equivalently
.
Clearly if
, then there is a
such that
. So
is a base for some topology on X. This topology is denoted by
.
More presicly
.
In [10] it is shown that
is Hausdorf, so if X is finite then we have the discreet topology, therefore interest- ing examples are when X is infinite. Also, if X is infinite then
should be infinite, other wise
, which implies also that the topology is the discrete topology.
Proposition 4. Let
be a semi-linear uniform space and
a subcollection of
satisfies,
i) For every
, there is a
such that
.
ii)
.
iii)
.
Then
is a semi-linear uniform space and
.
Proof. Since
is a subcollection of
, then
is a subcollection of
and i), iii), iv) in definition (1.1) are satisfied. Now for
ther exist
such that
so there is a
such that
So
is a semi-linear uniform space. Now
is clear. Let O be a nonempty open set in
then if
ther exist
such that
Let
such that
so
hence ![]()
Theorem 1. Let
be a semi-linear uniform space and
the topology on X indused by
, then.
i)
can be consider as asubset of ![]()
ii) Fore all
![]()
Proof. i). Let
, where
is the interior of U with respect to the topology
on
By Proposition 2.2,
is a semi-linear uniform space indusing the topology
on
Sine we deal with the toplogy
, we may replace
with
, so for all
for some
hence ![]()
ii) Is clear by definition of ![]()
In [11] , Tallafha gave some important properties of semi-linear uniform spaces, using the set valued map
and
.
Now we shall give more properties of semi-linear uniform spaces.
4. More Properties of Semi-Linear Uniform Spaces
Let
be a semi-linear uniform space, then
is a chain so there exist a well order set
such that
For
, then, ther exist
, such that
. That is, there exist
such that
. This implies, if
then
Hence ![]()
is bounded above by
. By Zorn’s Lemma
has a maximal element
. So
This copleet the proof of the following lemma.
Lemma 1. Let
be a semi-linear uniform space, then
where
is a well order
set, then for
, ther exist
such that ![]()
Remember that for all
the family of all entourages of the diagonal, V satisfies the following nice properties,
and
for all
. Let
and
be a subcollections of
, defined by
and
For all
by Co-
rollary 1.6, there exist
such that
. So we can define
, for an element
by.
Definition 5. For
and
. Define
by,
![]()
Clearly
and
for all
. But
need not be an element in
, evin if
. But we have.
Lemma 2. Let
be a semi-linear uniform space,
R is a well order set, and
.
Then there exist
, such that
and if
satisfied
, then ![]()
Proof. For
, there exist
such that
, so for every
satisfies
, we have
. Hence
is bounded above by
. Hence
be the maximal element of T.
Theorem 2. Let
, and
a subcolection of
. For
, we have
i)
.
ii) If
satisfies
, then
.
iii)
.
iv)
.
v)
.
vi)
.
Proof. i). Let
, then there exist
such that
![]()
So there exist
in
such that
and
Since
is a chain, there exist
such that
for all
So ![]()
ii) and iv), are clear by definition of ![]()
iii)
v)
, for all
so
. Conversly, Let
then, for all
there exist
such that
So there exist
in
such that
and ![]()
Since
is a chain, there exist
such that
for all
So ![]()
Let
and
replacing A by
or by
we have.
Corollary 2. For
where
is a semi-linear uniform spaces, we have,
i)
.
ii)
.
Corollary 3. Let
and
then
i)
.
ii) If
satisfies
then ![]()
iii)
.
iv)
.
v) If
satisfies
then ![]()
vi) ![]()
Also we have the follwing corollary.
Corollary 4. For
where
is a semi-linear uniform spaces, then
.
Corollary 5. For
,
if
then
is the largest element in Γ satisfies ![]()
Also by Definition 3.6, for
where
is a semi-linear uniform spaces, we have,
i)
.
ii) ![]()
Proposition 5. Let
be any distinct points in semi-linear uniform spaces
. Then,
![]()
Proof. Let
then
, then there exist
such that
So there exist
such that
but
is a chain implies the existence of
such that
for all
So
On the other hand by proposition 1.7,
. So for all
which implies ![]()
Definition 6. For
,
and
define
i) ![]()
and the greatest common devisor of
is 1.
ii)
.
iii)
.
Definition 7. Let
be any points in semi-linear uniform spaces
. For ![]()
, where
and the greatest common devisor of
is 1.
Using the a bove definition and Proposition 1.7, we have.
Proposition 6. Let
and
then ![]()
Proposition 7. Let
be any points in semi-linear uniform spaces
. If
then ![]()
Definition 8. Let
be an increasing sequence of positive rationals. If
then
is defined by ![]()
Proposition 8. Let
and
an increasin sequence of positive rationals. If
then
![]()
Proof. It is an immediate consequence of Proposition (7) and Definition (8).
5. Contractions
In [10] the definitions of converges and Cauchy are given. Now we shall discuss some topological properties of semi-linear uniform spaces. Since the semi-linear uniform space is a topological space then the continuity of a function is as in topology. The concept of uniform continuity is given by Wiels [4] , so we have:
Definition 9 [4] . Let
, then f is uniformly continuous if
such
that if
then ![]()
Clearly using our notation we have:
Proposition 9. Let
. Then f is uniformly continuous, if and only if
such that, for all
if
then ![]()
The following proposition shows that we may replace
by
in Proposition 2.2.
Proposition 10 [12] . Let
. Then f is uniformly continuous, if and only if ![]()
such that for all
if
then ![]()
In [10] , Tallafha gave an example of a space which was the semi-linear uniform space, but not metrizable. Till now, to define a function f that satisfies Lipschitz condition, or to be a contraction, it should be defined on a metric space to another metric space. The main idea of this paper is to define such concepts without metric spaces, and we just need a semi-linear uniform space, which is weaker as we mentioned before.
Definition 10 [1] . Let
then f satisfied Lipschitz condition if there exist
such
that
Moreover if
, then we call f a contraction.
Now we shall give a new definition of Lipschitz condition and contraction called r-Lipschitz condition and r-contraction.
Definition 11. Let
then f satisfied r-Lipschitz condition if there exists
such
that
Moreover if
then we call f a r-contraction.
Question. Let
be semi-linear uniform spaces and
what is the relation be- tween Lipschitz condition and r-Lipschitz condition, contraction, and r-contraction.
Remark 1 [12] . Let
be semi-linear uniform spaces, then if
then the topology indused by
is the descrete topology which is metrizable. Therefore we can asumme that ![]()
Question [1] [12] . Let
be a complete semi-linear uniform space. And
be a contraction. Does f has a unique fixed point?
Question. Let
be a complete semi-linear uniform space. And
be r-contraction. Does f has a unique fixed point?